Hailing from New Jersey, Kenwith had a long career on Broadway as well as in regional theater before turning to television. On Broadway, he was an assistant stage manager and performer in the original production of I Remember Mama, which ran for 713 performances between 1944 and 1946 and marked Marlon Brando's Broadway debut. Kenwith later produced the original Broadway production of Me and Molly, which ran for 156 performances in 1948 and featured David Opatoshu in the cast. After Broadway, Kenwith produced and directed 65 plays for Princeton University's McCarter Theater. Among the performers he worked with during this time was Lucille Ball, who later tapped Kenwith as a director on her show, Here's Lucy.

The majority of Kenwith's television directing credits are sitcoms, including Mary Tyler Moore, Sanford and Son, Good Times, Gimme a Break! and Bosom Buddies. He also directed and produced the entire first season of Diff'rent Strokes, which began airing in 1976. Other shows he directed and produced include One Day at a Time, The Facts of Life, and the short-lived NBC series Joe's World, which starred K Callan.

Kenwith was a personal friend of Lucille Ball, and asked her if he can direct an episode of Star Trek. The cast and crew were surprised on how well and easily Kenwith directed the episode, despite his inexperience with filmed drama series (his television credits mostly being sitcoms). (These Are the Voyages: TOS Season Three)